A
$d$dimensional
dual hyperoval
${\mathcal{S}}_{\sigma ,\varphi}^{d+1}$
inside
$PG\left(2d+1,2\right)$
($d\ge 2$) was constructed by
Yoshiara, for a generator
$\sigma $
of Gal$\left(GF\left(q\right)\u2215GF\left(2\right)\right)$ and an
opolynomial
$\varphi \left(X\right)$
of
$GF\left(q\right)\left[X\right]$
($q={2}^{d+1}$).
There, its automorphism group is determined and a criterion is given for
these dimensional dual hyperovals to be isomorphic, assuming that the map
$\varphi $ on
$GF\left(q\right)$ induced
by
$\varphi \left(X\right)$ lies in
Gal$\left(GF\left(q\right)\u2215GF\left(2\right)\right)$.
In this paper, we extend these results for a monomial opolynomial
$\varphi $. We show
that Aut$\left({\mathcal{S}}_{\sigma ,\varphi}^{d+1}\right)\cong G{L}_{3}\left(2\right)$ or
${Z}_{q1}.{Z}_{d+1}$ according
as
$d=2$ or
$d\ge 3$, if
$\varphi \left(X\right)$ is monomial but
$\varphi \notin $Gal$\left(GF\left(q\right)\u2215GF\left(2\right)\right)$.
In particular, a special member
$X\left(0\right)$
of
${\mathcal{S}}_{\sigma ,\varphi}^{d+1}$ is always fixed by
any automorphism of
${\mathcal{S}}_{\sigma ,\varphi}^{d+1}$.
Furthermore,
${\mathcal{S}}_{\sigma ,\varphi}^{d+1}\cong {\mathcal{S}}_{{\sigma}^{\prime},{\varphi}^{\prime}}^{d+1}$ if
and only if either
$\left(\sigma ,\varphi \right)=\left({\sigma}^{\prime},{\varphi}^{\prime}\right)$
or
$\sigma {\sigma}^{\prime}=\varphi {\varphi}^{\prime}=$id.
